PostGIS, ST_Intersects & LIMIT performance question

Question

I have a pretty standard table with one geometry column and a datetime column. Geom is indexed using gist and table is clustered + VACUUM ANALYZE. The Docker image is kartoza/postgis:12.0.

I consider the following query (bbox intersection):

SELECT product_id 
FROM products
WHERE 
    ST_Intersects(
        geom, 
        ST_MakeEnvelope(%s, %s, %s, %s, 4326)
    )
LIMIT 1000

I have noticed the following behavior: query runs < 10ms when the bbox is below 1 deg² (returning 1000 rows), but the performance degrades as the AOI surface increases (for example, 250ms if the bbox is 20 deg x 20 deg).

I would have expected the opposite as PostGIS is more likely to find quickly 1000 products intersecting a bbox as the bbox is larger.

So I guess the LIMIT clause is not quite effective in my query.

How can I correct this behavior, and implement a robust & fast pagination?